gloria marÍ beffa · resume. la theorie des invariants differentiels et les evolutions...

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GLORIA MARÍ BEFFAThe theory of differential invariants and KDVhamiltonian evolutionsBulletin de la S. M. F., tome 127, no 3 (1999), p. 363-391<http://www.numdam.org/item?id=BSMF_1999__127_3_363_0>

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Bull. Soc. math. France,

127, 1999, p. 363-391.

THE THEORY OF DIFFERENTIAL INVARIANTS

AND KDV HAMILTONIAN EVOLUTIONS

BY GLORIA MARI BEFFA (*)

ABSTRACT. — In this paper I prove that the second KdV Hamiltonian evolutionassociated to SL(n, R) can be view as the most general evolution of projectivecurves, invariant under the SL(n,R)-projective action on RP"'"1, provided that certainintegrability conditions are satisfied. This way, I establish a very close relationshipbetween the theory of geometrical invariance, and KdV Hamiltonian evolutions. Thisrelationship was conjectured in [4].

RESUME. — LA THEORIE DES INVARIANTS DIFFERENTIELS ET LES EVOLUTIONSHAMILTONIENNES DE KDV. — Dans cet article, je prouve que la seconde evolutionhamiltonienne de KdV, associee au groupe SL(n,R), peut etre consideree comme1'evolution la plus generale des courbes projectives qui sont invariantes par Factionprojective de SL(n, R) sur RP71"1, si une certaine condition d'integrabilite est satisfaite.Je mets alors en evidence une connection tres etroite entre la theorie d'invariancegeometrique et les evolutions hamiltoniennes de KdV. Cette relation a ete conjectureeen [4].

1. IntroductionConsider the following problem: Let ( / ) ( t ^ 0 ) € MP^"1 be a family

of projective curves. We ask the following question: is there a formuladescribing the most general evolution for 0 of the form

^=F(^^^...)

invariant under the projective action of SL(n,R) on RP72"1? Here

,/ / d0 / d<^0 =^e=^. ^ - d T

(*) Texte recu Ie 11 mars 1998, accepte Ie 17 septembre 1998.G. MARI BEFFA, Department of Mathematics, University of Wisconsin, Madison, WI53706 (USA). Email: [email protected] classification: 53Z05, 53 A 55.Keywords: differential invariants, projective action, invariant evolutions of projectivecurves, KdV Hamiltonian evolutions, AdIer-GePfand-Dikii Poisson brackets.

BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE 0037-9484/1999/363/$ 5.00© Societe mathematique de France

364 G. MARI BEFFA

The projective action of SL(n,R) on RP7'"1 is the one induced on W"1

by the usual action of SL(n, R) on R71 via the lift

ron—l •(i ,^) .As we showed in [4], such a formula can be found using the theory ofprojective differential invariance. In fact, one can prove that any evolutionof projectives curves which is invariant under SL(n,R) can always bewritten as

(1.1) rf>t=^

where 1 is a vector of differential invariants for the action and ji is aparticular (fixed) matrix of relative invariants^ whose explicit formula wasfound in [4]. Roughly speaking, if a group G acts on a manifold M, onecan define an action of the group on a given jet bundle J^ of order k,where J^ is the set of equivalence classes of submanifolds modulo bordercontact. This action, in coordinates looks like

GxJ^^jW,

(g,uK) •—^ (gu)K,for any differential subindex K of order less or equal to A:, and it is calledthe prolonged action. A differential invariant is a map

j: jW _, ̂

which is invariant under the prolonged action. A relative differentialinvariant is a map

j: jW _, ̂

whose value gets multiplied by a factor under the prolonged action. Thefactor is usually called the multiplier. In our particular case, their infini-tesimal definitions are given in the second part of section 2. Differentialinvariants and relative invariants are the tools one uses to describe inva-riant evolutions.

These two concepts belong to the theory of Klein geometries andgeometric invariants which had its high point last century before theappearance of Cartan's approach to differential geometry. It is alsoclosely related to equivalence problems. Namely, one poses the questionof equivalence of two geometrical objects under the action of a certaingroup, that is, when can one of those objects be taken to the other one

TOME 127 — 1999 — ?3

DIFFERENTIAL INVARIANTS AND KDV EVOLUTIONS 365

using a transformation belonging to the given group? For example, giventwo curves on the plane, when are they equivalent under an Euclideanmotion? or, when are they the same curve, up to parametrization? etc. Oneanswer can be given in terms of invariants, that is, expressions dependingon the objects under study and that do not change under the actionof the group. If two objects are to be equivalent, they must have thesame invariants. If these invariants are functions on some jet space (forexample, if they depend on the curve and its derivatives with respect tothe parameter), then they are called differential invariants. In the case ofcurves on the Euclidean plane under the action of the Euclidean group,the basic differential invariant is known to be the Euclidean curvature,and any other differential invariant will be a function of the curvatureand its derivatives. In the case of immersions

with SL(2,M) acting on MP1, the basic differential invariant is classicallyknown to be the Schwarzian derivative of 0,

^/. r 3/0'Ysw=~^~2(~^) '

Within the natural scope of the study of equivalence problems and theirinvariants lies also the description of invariant differential equations,symmetries, relative invariants, etc. For example, recently Olver et al. [12]used these ideas to characterize all scalar evolution equations invariantunder the action of a subgroup of the projective group in the plane, aproblem of interest in the theory of image processing. See Olver's book [11]for an account of the state of the subject.

A subject apparently unrelated to the Theory of differential invarianceis the subject of Hamiltonian structures of partial differential equations,integrability and, in general, of infinite dimensional Poisson structures.The so-called KdV Poisson brackets lie within this area. These bracketswere defined by Adier [1] in an attempt to generalize the bi-Hamiltoniancharacter of the Korteweg-deVries (KdV) equation and its integrability.He defined a family of second Hamiltonian structures with respect towhich the generalized higher-dimensional KdV equations could also bewritten as Hamiltonian systems. JacobFs identity for these brackets wasproved by GePfand and Dikii in [3]. These Poisson structures are calledsecond Hamiltonian KdV structures or Adier- Gel Jand- Dikii brackets^ andthey are defined on the manifold of smooth Lax operators. Since theoriginal definition of Adier was quite complicated and not very intuitive,

BULLETIN DE LA SOCIETE MATHEMATIQUE DE FRANCE

366 G. MARIBEFFA

alternative definitions have been subsequently offered by several authors,most notably by Kupershmidt and Wilson in [7], and by DrinfePd andSokolov in [2]. Once the second Hamiltonian structure was found, theintegrability of generalized KdV equations was established via the usualconstruction of a sequence of Hamiltonian structures with commutingHamiltonian operators. In this paper I will restrict to the case of theSL(n,R) AdIer-GePfand-Dikii bracket, although brackets have been givenfor other groups (DrinfePd and Sokolov described their definition forany semisimple Lie algebra). The second Hamiltonian Structure in thishierarchy of KdV brackets coincides with the usual second Poisson bracketfor the KdV equation, that is, the canonical Lie-Poisson bracket on thedual of the Virasoro algebra. This is the only instance in which thesecond KdV bracket is linear.

The relationship between Lax operators (scalar n-th order ODE's) andprojective curves was established by the classics and clearly described byWilczynski in [13]. More recently (see [12]) the topology of these curveswas used to identify one of the invariants of the symplectic leaves ofthe AdIer-GePfand-Dikii Poisson foliation. Some comments with respectto the role of projective curves in these brackets can be found in [14]and [7]. In [4] it was conjectured that the second KdV Hamiltonianevolution and the general evolution for projective curves (1.1) foundin [4] were, essentially, the same evolution under a 1-to-l (up to SL(n,]R)action) correspondence between Lax operators and projective curves. Theonly condition that needed to be imposed was that certain invariantcombination of the components of the invariant vector Z in (1.1) shouldbe integrable to define the gradient of certain Hamiltonian operator (onecan even describe the evolution so that both I and Hamiltonian coincideafter the identification).

In this paper I prove this conjecture. Namely, I prove that there existsan invariant matrix M, invertible, such that, ifHis the pseudo-differentialoperator associated to an operator 7^, and if

H=MIthen, whenever (f) evolves following (1.1) with general invariant vector Z,then their associated Lax operators (associated in the sense of [4] anddescribed again in the next chapter) will evolve following an AGD-evolution with Hamiltonian operator H. I also prove the conjecturedshape of M., namely, lower triangular along the transverse diagonal withones down the transverse diagonal and zeroes on the diagonal inmediatelybelow the transverse one. The proof is based on a manipulation ofWilson'santiplectic pair for the GL(?2, R)-AGD bracket and on the comparison ofthe resulting formulas with the invariant formulas (1.1).

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In chapter 2, I will briefly describe the 1-1 (up to SL(?2,IR) action)correspondence between the SL(n,R) AGD manifold and the manifoldof projective curves, and I will also describe the evolutions we wish torelate. In chapter 3, I will describe Wilson's theory of antiplectic pairsand, in particular, the antiplectic pair for the GL(?2,IR)-AGD bracket.In chapter 4,1 will adapt Wilson's formulas and finally give the proof of theMain Theorem. I will also prove that M. has the shape conjectured in [4].For more information or further references about either the evolutionsunder study or the correspondence between Lax operators and projectivecurves, please see [4].

2. Description of the manifolds and their evolutionsBefore describing evolutions, I will state the known parallelism between

the manifold of Lax operators and the manifold of projective curves.

Let An be the infinite-dimensional Frechet manifold of scalar differen-tial operators (or Lax operators) with T-periodic, smooth coefficients ofthe form

, , d71 d71-2 d(2J) L=~^JÛn-2^~^"ûlTeûQ'

The manifold An is called the SL(n, M) AdIer-GePfand-Dikii manifold, ormanifold of SL(n,IR)-Lax operators. (Notice that for much of the studybelow the periodicity condition can be omitted.) The case when Un-idoes not vanish is refered to as the GL(n, R) AGD manifold or manifoldof GL(n, M)-Lax operators.

Let ^L = (î) • • • j^n) be a solution curve associated to L, that is

L^=0, A ; = l , . . . , n ,

the Wronskian of whose components equals one. Notice that, since theWronskian of any solution curve is constant {un-i = 0), up to multi-plication by a matrix in SL(n,R), ̂ will be uniquely determined by itsWronskian being equal to 1. Now, due to the periodicity of the coefficientsof L, there exists a matrix ML G SL(n,R), called the monodromy of L,such that

^(<9 + T) = Mz^(6Q, for all 0 C M.

(ML is conjugate to the transposed of the Floquet matrix.) This same pro-perty is shared by the projective coordinates of its projection on RP71"1, asfar as we consider the action of SL(n, R) on the projective space. Observe


368 G. MARI BEFFA

that the monodromy is not completely determined by the operator L, butby its solution curves. Namely, if one chooses a different solution curve, itsmonodromy won't be equal to ML in general, but it will be the conjugateof ML by an element of GL(n, R). If we additionally ask the solution curveto have Wronskian equals 1, then ML is determined by the equation upto conjugation by an element of SL(n,R). Thus, to each Lax operator wecan associate a projective curve (the projectivisation of certain solutioncurve) whose monodromy is an element of SL(n,R). This curve is uniqueup to the projective action of SL(?2,IR).

Conversely, let Cn be the space of curves

J,. TQ) ___, 7D) "p^ — 1(p . M ——>• IKIr

with the following nondegeneracy condition: the determinant

^"-1)/,("-!)°2

^(»-1)Pn-1

€ ^

4>2 ^

I I

n-1/n-1

must be positive. (This is equivalent to the Wronskian of the componentsof (1,0) being positive; for example, the curve would be convex and right-hand oriented in the case n = 3.) Assume also that the elements of Cnsatisfy a monodromy property:

(2.2) 0(<9 + T) = (M • 0)((9), for all 0 6

for some M e SL(7i,R). Here M • (f) represents the projective action ofSL(n,R) onW"1.

To each (j) one can associate a differential operator of the form (2.1)in the following manner: We lift (j) to a unique curve on R71 so that theWronskian of the components of the lifted curve equals 1. There is aunique choice /(^)(1,<^), namely when

/ = W(l, 0i , . . . . (^-i)-^ = W(^.... <^_i)-* = W^,

where (f) = (0i , . . . , (f)n-i) and W represents the Wronskian. It is not veryhard to see that the coordinate functions of the lifted curve are solutionsof a unique differential equation. The equation for the unknown y is of

TOME 127 — 1999 — ?3


the form

(2.3)

yW (f^)W

V^^ Wo)^

/00

Wn-l)^

(f^n-l)^

f^n-1

= y^ + Un^y^'^ + • • • + u^y' + uo2/ = 0,

where 0o = 1- O116 can easily see (cf. [4]) that the monodromy property(2.2) results on the coefficients i^, i = 0,1, . . . , n — 2 being T-periodic, sothat this equation corresponds to a Lax operator of the form (2.1).

It is known (see [13]) that the coefficients of this Lax operator formthe so-called generating set of differential invariants for projective curvesunder the projective action of SL(n, R). Namely, if a function J, dependingon (f) and its derivatives, is invariant under the prolonged projective actionof SL(n, IR), then, necessarily I is a function of the coefficients u^s and theirderivatives. The u^s can thus be called the "projective curvatures" of thecurve (/). I will go back to this point in the description of the evolution ofcurves. For more information see [13] and [4].

EXAMPLE. — In the case of immersions (/): R —^ RP1, with SL(2,R)acting on RP1, the Wronskian W^ equals <j)'. Thus, the lift ^ is given by(^i, ̂ 2) = (0'~2 ^ ^'"^(f^). If we write down the associated Lax operator forthis particular example, we obtain

(^y o^n=y"+^S((f,)y,W-.y^ (^/-^y

^-4where, again, S((f)) = <y" 1 ^ > ' — |(<^//1^/)2 is the classical Schwarzianderivative of (^, the basic differential invariant or projective curvature.

The above description establishes a 1-1 (up to SL(n,R)) correspon-dence between An and Cn' Next I will describe two different evolutions,one in each manifold.

The AdIer-Gel'fand-Dikii bracket, or the evolution on An'Given a linear functional Ji on An-, one can associate to it a pseudo-

differential operator symbol of the form

(2.4) H=^hi9~\ <9=d

~d0


370 G. MARI BEFFA

such thatH(L)= [ ves(HL)d0,

J s 1

where 'res' selects the coefficient of 9~1 and is called the residue ofthe pseudo-differential operator (see [1] or [3]). The coefficients h^i = 1, . . . , n - 1 are combinations of the coefficients of the gradient QH/Quand their derivatives; the coefficient hn will be determined below.

To any H we can associate a (Hamiltonian) vector field VH defined as

VH(L)=(LH)^L-L{HL)^

where by ( )+ we denote the non-negative (or differential) part of theoperator. The vector field VH defines a bracket, namely

(2.5) {H^}(L)= [ res(l^(L)F)d0,J s 1

which turns out to be a Poisson bracket with associated Hamiltonianevolutions given by

(2.6) Lt = Vn{L)

cf. [I], [3] or [9]. The coefficient hn of the operator H is fixed, and itsvalue easily found, so as to make the vector field VH tangent to themanifold An (that is, so that both sides of (2.6) have the n - 1 term equalzero). These are called the SL(n,M) AdIer-GePfand-Dikii evolutions. AsI explained in the introduction, the original definition of the bracket wasgiven by Adier [1] in an attempt to make generalized KdV equations bi-Hamiltonian systems. GePfand and Dikii proved JacobPs identity in [3].In the case n = 2, this bracket coincides with the Lie-Poisson structureon the dual of the Virasoro algebra. Two other equivalent definitions ofthe original bracket were found in [6] and in [2]. I will describe briefly thedefinition in [6] in the chapter where I prove the Main Theorem.

Invariant evolutions of protective curves, or the evolution on Cn.On Cn we are interested in evolutions of the form

(2.7) ^ = F(^ 0', ̂ / / , . . .) , 0: R2 —— Rp—i,

such that whenever (f)(0,t) is a solution of (2.7) so is (M • (f))(0,t), for allM € SL(n,M). That is, evolutions of curves on RP71"1, invariant underthe projective action of SL(n,R).

TOME 127 — 1999 — ?3


One can see that, if the initial condition has the monodromy property,the entire flow does and in fact the monodromy is preserved, as far asthe solution is unique. Indeed, (2.7) is invariant under translations of theindependent variable 0. Hence, if the initial condition 0( • , 0) of (2.7) hasa matrix M € SL(n, M) as monodromy, and we consider a different curvein the flow (f)( • , t), we have that (f)(0 — T, t) is also a solution. If (2.7) isSL(n, R)-invariant, M'(f)(0—T, t) will also be a solution of (2.7). Applyinguniqueness of solutions of (2.7) (whenever possible),

M -(f){e-T,t) =(/)(0,t),

so that ( f ) ( ' ,^) has the same monodromy as 0( - ,0 ) . If there is nouniqueness of solutions, both Hamiltonian and invariant evolutions areobviously much more complicated; I won't deal with those cases in thispaper.

In [4] we found the explicit formula for the most general form of evo-lution (2.7). It could be described as follows: First of all, the infinitesimalgenerators of the projective SL(n, R) action on RP71"1 are easily found tobe the following vector fields on R x M x W"1:

r\ r\ 1T' r\(2.8)^=—. v^j=(l)^^T~^ w! = ̂ y^^'^r^ i^j^^-i-dcpi d(j)j ^ d(j)j

n—in-lS ''Iz^f ̂i=l

If -y = ̂ y^^^^— is a vertical vector field, its prolongation prv

(see [9]) is the vector field defined on the jet space by

8n-l

(2.9) ^=^Y.WD^-j^l i=l ^W Vi )k>0

where (û) = Qî, D is the total derivative operator with respect to 0

D=9+Y.Y,<P(,j+l)9/-^ Z - ^ ' 1 QJL(.0)J'Ô 2=1

and

n-l 9(2.10) Dt=9t+^^9^)^^î ; y)jÔ i=l ^Vi


372 G. MARI BEFFA

The prolongation of an infinitesimal generator is indeed the infinitesimalgenerator of the prolonged action on the jet space. In our case, it reducesto the vector field

(2.11) prv=v+^^{D^i)^9^j>l i=l

defined on the infinite-dimensional jet space

TOO —— TOO /TTp TOTir7'1"^J = J ^M, Mir )

with local coordinates 0, (p^ where 1 < i <^ n— 1 and j > 0 (to be correct,we should in fact restrict ourselves to the jet space J^\ for some order kas large as necessary; but for simplicity I will work on J^).

DEFINITION 2.1.a) We will say that J is a differential invariant for the SL(n,R)-

projective action if

PTV(I) = 0 for all v € sl(n,IR).

This is indeed the infinitesimal version of the definition we gave in theintroduction: I is invariant under the prolonged action.

b) (Infinitesimal definition also) Let

n—l ^

=y^((9,^)—— esl(n.R)^ ^

v =

be an infinitesimal generator. We will say that F is a relative vectordifferential invariant of the Lie algebra sl(n,M) given by (2.8), whoseassociated weight is the matrix 9rj/9(/) whenever

(2.12) pr^(F)=^F,

for all v e sl(n, R), where Orj/Qcf) is the (n - 1) x (n - 1) matrix with (ij)entry 9rji/9(f)j.

It is not hard to see (cf. [4]) that (2.7) is invariant if, and only if Fis a relative vector differential invariant of the Lie algebra sl(n, R) withweight Qrj/Qcf).

TOME 127 — 1999 — ?3


THEOREM 2.2 (see [4]). — The most general solution F of (2.12) is ofthe form

F=^.

where the {n — 1) x (n — 1) matrix

/i=(^2... ^"-1)is any matrix with non-vanishing determinant and whose columns ^ areparticular solutions of (2.12), and where

T- =(Wi1

is an arbitrary absolute (vector) differential invariant of the algebra (2.8),i.e. a solution of

prv(Ii) =0, W e 5[(n), i = 1 , . . . ,n - 1.

The problem is thus splitted into two parts: first find the invariants,second find the matrix fi.

The first part was already solved by Wilczynski in [13]. He provedthat, in the case of projective curves, a set of generating basic differentialinvariants are the coefficients of the differential operator associated to (f)as in (2.3). That is, any differential invariant Ij will have to be a functionof the coefficients Uz, i = 0 , . . . ,n — 2 and their derivatives with respectto 0.

In [4] we solved the second part, finding an explicit expression for aregular matrix of relative differential invariants, ^, which I describe below.

DEFINITION 2.3.a) For % i , . . . ,%fc > 0 and 1 <, k <, n — 1, let us denote

Wi

^1)

^

^

^^2)

^

... ^-. ^2)

.:: ^and

W^ = Wi2...fc.

b) We define the homogeneous variables ^i^...^ by

(1^1^2.••ik =W,' Z l Z 2 - - - Z k

W1"


374 G. MARI BEFFA

c) For k = 1,2,. . . ,n the variables q^ are defined as follows:

^=9l2...fc...»'

where the notation k means that the index k is to be omitted.

THEOREM 2.4. — An invertible matrix /j, of relative invariants withweight Qr]/9<p is given by a matrix of the form

W^(Id+A)

where W^ is the transposed of the Wronskian matrix of <j) defined as

( ^ ^ . . . ^-i \

(2.13) W^ =

P'n-,

^ ^ ... <_l

U^ ^-1) ... ^)

and where A = (a^) is defined by

[ (-Ip-1^)I \Z / n— Q-i-i • r •

(2.14) ai = { ( n ^ ^ J+ ^ ^ < ̂| \ J - i }

0 if i > j .

As an immediate consequence one obtains:

COROLLARY 2.5. — The most general equation for the evolution ofcurves on RP71"1 which is invariant under the projective action ofSL(n, R)is given by

(2.15) ^=H^(Id+A)Z,

where YV^ and A are given by (2.13) and (2.14), and where I is any vectordifferential invariant for the action.

Before finishing the section I will like to include an immediate corollaryto Theorem 2.2 which will be of use later on.

TOME 127 — 1999 — ?3


COROLLARY 2.6. — Let fJi and /2 be two nondegenerate matrices whosecolumns are relative vector differential invariants, with associated weightsgiven as in (2.12). Then, there exists an invariant and nondegeneratematrix M. such that

^ = JIM.

After defining the two parallel evolutions, and describing some of thetheory of differential invariance, I would like to show that these areessentially the same evolution. This implies that certain relationship willhave to be found between I and H. Notice that both of them are functionsof the coefficients ui, i = 0 , . . . ,n - 2, and their derivatives. In fact/itwas conjecture in [4] that both equations (2.6) and (2.15) are equivalentwhenever

(2.16)

where

m_6u

=TMI,

(2.17) T=

t 1

992

93

gn-3

^gn-2

(^(I

... ("33)^fn-2Vn-3

0

1

)92

^gn-3

0

0

1

(^

/n-2V 3

0

0

0

1

/n-3V 2 )92

)93

0

0

0

0

(n^(n,2)^

... 0\

... 0

... 0

... 0

1 0

("r2)^/and M. is a certain lower triangular matrix of the form

(2.18) M=

/O00

0

\1

1

0

000

0^-3'^n^

0

01

-m 7l-4'^71-4

yy^-477t'n-3

0

1

0

... r<-4

10

m\

m^-3

\

)

whose matrix elements m\ are all functions of the coefficients HI and theirderivatives. On the other hand, if

H^^hkQ^k=l


376 G. MARI BEFFA

is the pseudo-differential operator associated to 7^, ( / ^ i , . . . , hn-i) is easilyseen to be related to the gradient of 7i through the matrix T, exactly thesame way M.T is conjectured to be in (2.16), namely,

^f"' vsu ^jThat is, the evolutions will be equivalent provided that

( hl \MI=( : ,^n-l7

that is, provided that certain linear combination of 1 with differentialinvariant coefficients coincides with the coefficients ( / i i , . . . , hn-i) of thepseudo-differential operator H defining the evolution of u. The necessaryand sufficient condition for this to be true is, of course, for TM.I to haveself-adjoint Frechet derivative with respect to u. Notice the slight changein the shape of T and M. as opposed to the ones conjectured in [4]. Thechange is due to the fact that T was compared in [4] to (^n-i, . . . , h^and here I have turned it into the more natural order ( / i i , . . . , hn-iY^ .

Before I specify all the details of the proof in Section 4, I need to givea brief description of what is called the theory of antiplectic pairs, whichwas introduced by Wilson in [14], [15] and [16], and, in particular, of theantiplectic pair for the GL(TI,]R)-AGD bracket. In the next section I willkeep much of the notation as in [14], but I will try to avoid any confusionwith the notation I have used up to this point.

3. Antiplectic pairsLet (A, 9) be the differential field of differential rational functions on

independent variables ^05^1? • • • ̂ n-i' That is, A is the field of ratio-nal functions on the infinite-dimensional jet space J00^,!^) with coor-dinates ^o? ^15 • • • 5 ^n-i-> ^d 9 is the unique derivation such that9 S , ' == ^ ' . I will assume familiarity with concepts such as Frechetderivatives of differential operators on A, tensors, etc^ such as we didin [4]. For more information see [14] or [9].

Consider the action of GL(n,R) on A induced the usual way by theaction of GL(n, R) on IT. That is, if MS, indicates the action of GL(n, R)on W given by matrix multiplication, ^ € R71, then M^ = (M^)^,what we have called the prolonged action. Let (B, 9) be the differential

TOME 127 — 1999 — N° 3


field of invariants of the action. As it happened in the SL(n,M) case, Bis well known to be generated by the coefficients of the GL(n,R)-Laxoperator (2.1), that is, the case where Un-i i=- 0; B is a differential fieldon the independent variables UQ, HI, . . . , u^-i, related to ^ analogously asin (2.3), with associated derivation 9.

Let Â and ^5 be the modules of differentials. That is, they are freeA[(9]-module with^basis {d^}, and free B[<9]-module with basis {dn,},respectively. Let f2a be the module flp with scalars extended to A[9\.If we impose that the action of GL(n,M) conmutes with d, the GL(n,R)prolonged action can be extended to ÎA the obvious way. Consider also fTto be the dual ^-module to f2, that is, ^* = Hom(^, R) where R = A[9]is the algebra of differential operators with coefficients in A. Analogouslydefine ̂ , ̂ and %.

Next, let us describe the general Hamiltonian formalism for evolutionequations.

DEFINITION 3.1. — A 2-form A e Â ^ Â, A = ̂ ^ 0 ̂ ,, is calledspecial if: l

1) A is GL(n,]R)-invariant;2) the homomorphism A^: f^^ ^- f^ defined as

^(^ ̂ ^^^z)^

is injective and its image is ^5.

Jf A is special, we obtain a factorization of A^ through the inclusioni: ÎB ^ Â; namely,

(3.1) A^-^^-^

where a is invertible (an isomorphism). By the way, notice that, withrespect to the basis {d^} and {dui} for Â and â and their dual basis{d^,*} and {dn,*} for ̂ and ̂ respectively, the matrix associated tothe inclusion map has as (ij) entry Dui/D^j, the Frechet derivative ofn,with respect to ̂ . That is, if we denote by P the matrix of the inclusionmap %: Q.B -^ Â, then V is the Frechet Jacobian of u with respect to $.

Now, consider the map

(3.2) A: ^B ̂ Â ̂ ^B'


378 G. MARI BEFFA

Since A is GL(n,R)-invariant, the map (3.2) must also be invariant.Therefore, it must come from a homomorphism

£*:^B—%by perhaps extensions ofscalars. The map C^ corresponds to a skew tensor£e %(g)^g.

Let S = (sij) be the matrix of a~1 in the basis {ck^}, {d^} and{d^*} that were fixed before. In the definitions that follows, let * denotethe formal adjoint of an operator. Then the matrix of A (or of A^ if oneprefers it), abusing the notation, is given by

A = S^V = -P*(5'*)-1 = (A^-).Also, in the basis {di^}, {d^*} and {d^*}, the matrix of£ (or £^) is

t == SV = -PS'* = (£ij).In this situation, and under all these conditions, there exists a unique

evolutionary derivation of B such that

(3.3) <^=-E^-3 J

Furthermore, there also exists a GL(n,R)-equivariant derivation 9n of Agiven by

(3.4) ^=E4^3 J

whose restriction to B is (3.3).DEFINITION 3.2. — The pair (A,^) is called antiplectic whenever £ is

Hamiltonian, that is, whenever the derivation given in (3.3) holds9{H,G} = [9H,9G\

if it makes sense, where {H\G} = QuG.It can be proved that if A is a closed form, then £ is Hamiltonian.In [14], Wilson constructs an antiplectic pair for the GL(n,R) KdV

evolution, restricting himself to a particular symplectic leaf of the Poissonfoliation. Even though at the beginning of the paper he imposes two extraconditions which are not included here, namely that the functions {^}must be periodic, and the field C rather than R, the construction above,and the one that follows, can be carried out without such assumptions.(In fact, he seems to drop the periodicity assumption until he places theantiplectic pair within the DrinfePd and Sokolov formulation. In that case,for algebraic and invertibility reasons, the solutions of the Lax operatorsneed to be periodic — he is restricting to one symplectic leaf of the Poissonmanifold with identity monodromy.)

TOME 127 — 1999 — ?3


Construction of Wilson's GL(n,R)-KdV antiplectic pair.Consider the factorization of L into first order factors

(3.5) L = (9 + 2/,_i). . . (0 + y,)(9 + yo)

such that for each i = 0 ,1, . . . , n - 1 the set {$o , . . . . <^} is a basis for thekernel of the operator (<9+^). • • (9+^/i)(9+?/o). Let ̂ be the Wronskianof{ô , . . . , ^} . Define

^Pz = ————

î-1

and fix c^_i = 1.As before, let f^ be the free module of differentials with base dp,, and

let f^ be ^ with an extension of scalars to the algebra A[<9], for anyi = 0 ,1, . . . , n - 1. Define the 2-forms

A^ = -pi^dpi (g) Qp^dp^ i = 0 ,1 , . . . ,n - 1,

and let ij: f^ —> ^^ be the natural inclusions j = 0,1, . . . , n — 1.THEOREM 3.3 (see [14]). — The 2-form

(3.6) A, = zoA^ + ziA^ + ... + z,-iA^-1)

Z5 special under the action ofGL(n,R) and forms an antiplectic pair withthe GL(n,R)-AGD Poisson tensor.

For more information about this very inspiring construction, pleasesee [14].

I am most interested in one of the explicit formulas for the matrix 5'~1

defining the map a in (3.1). More precisely, I am interested in formula (5.9)in [14], which can be described as follows: Let

n-lL*=(-9)n+^^

1=1

be the dual operator to L. There exists a nondegenerate canonical pairingbetween the kernels of L and L* given by the so-called bilinear concomi-tant. It is defined as

(^)L = EE^1)^"1^î=l j=0

(see [5] for more information). Let {rji} be the dual to {^} with respectto this pairing. Also, fix the basis {dui} in â and {d^,*} in ̂ .


380 G. MARI BEFFA

PROPOSITION 3.4 (see [14]). — If we denote by u the coefficients of L*,the matrix S~1 associated to \n as in (3.6) is given by

(3.7) S-1 = H^*C7*

where Du/Du = (7*, and where W^ is the Wronskian matrix of rj, thatis, )% = (y^) where y^ = rj^ for ij = 0,1, . . . ,n - 1.

We now have most of the machinery I need to attack our problem, sowe finally go into out last section.

4. The equivalence of evolutions on An and CnIn this section, I will prove our Main Theorem, the equivalence of the

two evolutions described in Section 2, and the description of M. But let'sfirst give a quick definition and let's set in place some preliminaries.

DEFINITION 4.1. — We will say that a vector funtion H = (h,) comesfrom a gradient whenever there exists an operator H: AGD -^ R suchthat its associated pseudodifferential operator, in the sense of (2.4) is

X=^hi9-^

i=l

That is, a vector function will come from a gradient whenever certaincombinations of its entries and their derivatives satisfy the correspondingintegrability condition so that they can be the gradient of an operatordefined on the AGD manifold.

The next thing to achieve is the rewriting of formula (3.4) in theSL(n,]R) case, and to show that it induces an evolution on the projecti-visation of ^. This evolution will be invariant under the projective actionof SL(n, M). The proof of the Theorem will finally come from the compa-rison of the formula we will obtain and (2.15).

First of all let us consider the GL(n, M)-AGD evolutions (3.3) restrictedto the submanifold Un-i = 0, with the added condition 61-i/6un-i = 0.I claim that, for Hamiltonians independent of Un-i, the submanifoldUn-i = 0 is left invariant by the flow of evolution (3.3), and so, there existsan induced evolution on such submanifold. Furthermore, I claim that suchan evolution is the SL(n,R)-AGD evolution. This fact can be seen inmany different ways, but the easiest might be to follow Kupershmidt andWilson's definition of the AGD evolution given in [6]. Before stating thetheorem that describes the bracket, I need some definitions.

TOME 127 — 1999 — ?3


Consider y defined as in (3.5) and let v be defined through the relation

(4.1) yk = vo + r^i + r2^ + — + r^-^Vn-^

0 < k < n — 1, where r = e~^~. Kupershmidt and Wilson [6] called Vz the"modified" variables. In their paper, they proved the following theorem(which I have somehow rephrased).

THEOREM 4.2 (see [6]). — Consider the following two evolutions for themodified variables v:

(4.2) QfVi == --Oxn-z, i = 1 ,2 , . . . ,n- 1n

(4.3) 9tVo=--9xo, QfVi == -~9xn-i, i = 1 ,2 , . . . ,n - 1n n

where (XQ^ ... ̂ Xn-i) is the 1-form defined by the gradient of a Hamilto-nian operator 7^, and where XQ corresponds exactly to 6H/6vQ. If u and vare related by (3.5) and (4.1), then u will evolve following the SL(?2,R)AGD evolution whenever v evolves following (4.2), and the GL(n,R) AGDevolution whenever v follows (4.3). The Hamiltonian will be the same^ butevaluated either on u or ^, depending on which evolution we consider.

That is, in modified variables the SL(n,]R) and the GL(n,R) AGDbrackets are given by (4.2) and by (4.3), respectively. Clearly, the subma-nifold VQ = 0 is invariant under the later flow (4.3), provided that XQ = 0,and the evolution induced on this manifold is the former (4.2). Finally,noticed that Un-i = nvo so that both submanifold VQ = 0 and Un-i = 0coincide. Thus, the restriction of (3.3) to Un-i = 0 is the SL(n,]R)-AGDHamiltonian evolution.

In second place we inquire: how does the restriction Un-i = 0 lookwhen carried over to the $ evolution (3.4)?

The condition Un-i == 0 imposes a condition on ^, namely, given L suchthat Un-i = 0, any set of independent elements in the kernel ̂ • • • ? ̂ n-i,will have constant Wronskian, that is, independent of the parameter Q.Call the Wronskian W^ = detH^. Furthermore, if they evolve usingevolution (3.4), we get the following claim.

PROPOSITION 4.3.—In the SL(n,R) case {that is^ whenever Un-i = 0),W^ is independent oft and 0 along the solutions of (3A).

Proof. — Indeed, if $ evolves following (3.4), due to the antiplecticrelation between A and £, u will evolve following the SL(n,R)-AGD


382 G. MARI BEFFA

evolution. It is known that the AGD evolution preserves the SL(n,IR)conjugation class of any of the monodromies associated to L. (It preservesthe GL(n,M) class in the GL(n,R) case.) In fact, the conjugation classof the monodromy is one of the invariants of the symplectic leaves ofthe AGD Poisson bracket, the so called continuous invariant. The secondinvariant is discrete (it does not change locally) and, as I remarked earlier,it depends on the topology of the projective curves (see [12] for thedescription of the invariants). That is, the determinant of the monodromyis unchanged along flows of the ^-evolution for <^. This determinantcoincides with the Wronskian of ^ at 0 = T, the period. []

Thus, equation (3.3) restricted to Un-i = 0 is the restriction of theevolution (3.4) to the ring B generated by n,, i = 0 , . . . , n - 2, wheneverone adds the extra condition to (3.4) of the Wronskian of^ being constant.Notice that this is a restriction on the set of initial conditions that areallowed in (3.4), rather than on the equation itself. The ^ evolution is, ofcourse, invariant under the real action of SL(n,M) (action on IR"), sinceit is invariant under the real action of GL(n,IR).

I must say that a SL(n, R)-invariant evolution on ^ induced by theAGD bracket was found in [7]; even though the approach used in thatpaper was quite nice, it was completely different from the one used byWilson (and the one used in this paper) and it is not obvious thatboth formulas coincide. In that paper the author also claims that itsuffices to take the restriction of the ^-evolution to the field generatedby ^ = <^/<^ i = l , . . . , n - 1 in order to obtain the so-called Ur-KdV equations, which will also be SL(n,R)-invariant. In fact, if we writethe evolution for the proportions (^ induced by (3.4), these equationswill not be homogeneous. One needs to futher substitute the relationship

^ = ^0 n(^ i = 1,... ,n - 1 and $o = W^ to obtain an evolutionon (/). These relationships exist only under the condition W^ = 1, and notsimply constant.

Finally, I will describe the evolution induced on 0, on the projectivisa-tion of the solution curve, by the evolution (3.4) in the SL(n,M) case.

Consider (3.4) restricted to curves with the condition W^ = 1 (we aresimply restricting further the possible initial conditions). Then, it is nothard to see that, if <^ = ^/^o is the projectivisation of ^

W,0 =

4>[ i o ... o01 0'1 ... 0n-l

^-1) ... ^1)/ j.(^-l) An-1)Z^n-l 01 / . . . <^_i /

TOME 127 — 1999 — N° 3


-so"

sosi

SO^.(n-1)sosi

n-1)

=^0

.(n-1)Sn—1

^ ^/I Sn-1 Sn-1

Therefore, the evolution (3.4) restricts to (j) as

(4.4) / s.<^= -T

so(so).

so

where the right hand side depends on (f) after the substitution

s.-^"^ % = ! , . . . , n-1 and ^^H^71.

LEMMA 4.4. — Let (3.4) 6e an evolution on ^, with W^ = 1, invariantunder the real action ofSL(n,R). Let (4.4) be the induced evolution on theprojectivisation of ̂ . Then, (4.4) is invariant under the projective actionofSL(n,R) onR?^.

Proof. — The proof is very simple of course. We need to show that, if weU>n—ldenote by M - 0 the projective action of SL(n, R) on (that is, the

projectivisation of MQ), then M • 0 is a solution of (4.4) whenever (f) is asolution itself. But notice that M(1) = ̂ M, and so, the projectivisationof M(^) coincides with the projectivisation of M^. If (f) is a solution of(4.4), it is clear that ^ is a solution of (3.4); because of the invariance of(3.4), M^ will also be a solution of (3.4) and its Wronskian will be 1 sinceM € SL(n, R). Hence, M • 0 will be a solution of (4.4). Q

After these three steps I proceed now to state and prove the MainTheorem, conjectured in [4].

THEOREM 4.5. — Let (l)(t,0) be a solution curve of the evolution

(4.5) ^=W^(Id+A)Z

defined in (2.15). Let u(t,0) be the vector of associated basic SL(n,R)-differential invariants defined in (2.3). Then, there exists an invariantand invertible matrix M. such that, if H = M.T and if H comes froma gradient, then u(t,6) evolves following the SL(n,R)-AGD Hamiltonianevolution

,6HUt=i——

buwhere H is the operator associated to H.


384 G. MARIBEFFA

Proof. — The proof is based on the analysis of (4.4) and its comparisonto (2.15). Recall that the ^-evolution can be written as

t Q*^St = ^ -7——?ou

where 6'* = W^G is denned in (3.7), that is, where (7* = Du/Du andwhere )% is the Wronskian matrix of 77. Let's call )%-1 = (^) andC = (cij). Using this expression for ^ we can show that

1) In general, the revolution (4.4) does not depend on 6H/6un-\, evenif (3.3) was dependent of it.

2) In expression (4.4), the coefficients of Cn-ij, j = 1, 2 , . . . , n - 1 arealways zero.

Indeed this is true. The coefficient ofSH/Sun-i in (3.4) is (-l)71"1^-!since, clearly, C is lower triangular and c^-in-i = (-I)""1. Now, it isknown (see [14]) that, if two basis are dual with respect to the bilinearconcomitant, then any of the basis form the last column of the inverse ofthe Wronskian matrix of the other basis. Thus, the last column of )%~1

^s (^0^1,. ..,^-1)^ Therefore g^-i = ̂ . Finally, the 6Zi/6un-i termin the 0-evolution has coefficient

l(-l)n-l^-^-l)n-l^=0.so so

Also, the coefficient of the entry Cn-ij in evolution (3.4) is given by9in-i = sz- Hence, as before, the coefficient of Cn-ij in evolution (4.4) isgiven by

^-^^1=^^.^so $o so so

In view of 1) and 2), we can deduce that the projective evolution (4.4)can in turn be written as

(4.6) <f>t=BC6'Hbu

where

M (M/ew'6u :

\6n/6un,^

TOME 127 — 1999 — ?3


C is the upper (n - 1) x (n - 1) block of C,

C=

0 \C \

0^nl • • • Cym—l C"nn /

and where, if I denote by G the last n - 1 rows of )%~1 and by Go itsfirst row, then B can be written as

(4.7) B - ( G - ^ G o ) .So

The final step is to realize what C = (c^) is. It is easy to see that, inour notation,

^-(-ir1^)^ ^=l ,2 , . . . ,n - l .

That is, CR = T, where R = (r^-) is given by r^ = (-lp-i^\(6 represents the Delta of Kronecker), and where T is defined in (2.17).Up to R, a matrix changing the sign of every other row, the matrix Du/Duand T such that 61-L/6u = TH are the same. (Notice that T = CRand C2 = Id by definition. Hence T~1 = RTR.)

Finally, equation (4.4) can be written as

(4.8) ^=BTR8-H=BRH.ou

This evolution is SL(n, IR)-projectively invariant for any gradient vector61-L/6u. The vector TR6H/6u = RH depends on the u's and theirderivatives. It is, therefore, an invariant vector. Now, in [4] we showed thatsuch an evolution is invariant if and only if BRH is a relative invariantfor the action with weights as in (2.12), and this should be true for anyinvariant vector H coming from a gradient. But if H is invariant we have

n-l

E-'i=l

that, if v e sl(n,R) is an infinitesimal generator v = ^ ^((9^^)—/•>v^) 9^,

then,

prv(BRH) = prv(BR)H = ^BRH.d(p

Running H through an appropiate independent family of vectors weobtain that BR is indeed a relative invariant with the necessary weights,that is

prv(BR) = J-BR^

for any infinitesimal generator v = ̂ rji —.i


386 G. MARI BEFFA

The end of the proof of our theorem is now clear. Using the resultof Corollary 2.6 we can deduce the existence of an invertible matrix ofdifferential invariants M. such that

(4.9) ^(ld+A)=BRM.

Therefore, if T^SH/Su = H = MI, evolutions (2.15) and (4.4) areidentical. Finally, the basic differential invariants u, which coincide withthe coefficients of the Lax operator L, evolve following the SL(?2,]R)-AGDHamiltonian evolution (3.3). []

A short comment on the meaning of this theorem. From Theorem 2.2we see that the matrix j£ = WÎd+A)^"1 is a relative invariant ofthe SL(77,,IR) action, since M. is a nondegenerate matrix of differentialinvariants. Thus, from Theorem 2.2, the most general invariant evolutionof projective curves can also be written as

(4.10) ^=/^

where T is any general vector of differential invariants. If 1 comes from agradient, then (4.10) can be view as an alternative definition for the AGDbracket, written in projective coordinates. They are the same evolutionwhenever one identifies Z, the invariant vector, with the pseudodifferentialoperator associated to a functional 7i, the Hamiltonian.

The last part of this chapter is dedicated to show that M. is indeedof the form (2.18). This is the result of the following two lemmas andtheir projective counterparts. As before, I will call W^ the Wronskian of ând H^ its Wronskian matrix. Analogous notation is used for rj.

LEMMA 4.6. — Let {î} and {rjz} be basis for kernel(L) and kernel(L*),respectively, and assume they are dual with respect to the bilinear conco-mitant. Then

W ^ M r 0 if k < n - r - l ,^ ll \(-1Y if fe=n-r-l,r=0,l,...,n-l.

Proof. — This lemma is a simple consequence of the relationship

v- Ak) _ f 0 if 0<k<n-l,^ ^" f l if k=n-\

which holds since rj conforms the last column of the inverse of theWronskian matrix of ^. The result of the lemma is obtained by simplyapplying Leibniz's rule.

TOME 127 — 1999 — N° 3


LEMMA 4.7. — IfW^ = 1, then W^ = (-I)7711, where mi = jn(n - 1).

Proof. — The previous lemma implies that W^)% is triangular alongthe transverse diagonal. Hence, W^ is the product of entries in thetransverse diagonal, namely (-I)7711, mi as in the statement. Q

LEMMA 4.8. — Let {(/),} be the projectivisation of {^}, and let {^} besuch that W^=l. Let {rji} be as in Lemma 4.6. Then

V-^) (.) f ° z f 0 < k < n - r - l ^y ^ ̂ ) = { iU \(-lYWy if A;=n-l-r,r=0,l, . . . ,n-2.

Proo/. — The proof is straightforward from Lemma 4.6. Namely, if0 < k < n - r - l , then

E^^Ê^)^1=1 j=i so

=ElE(')^-l)(fc-^(\(r)=oj=0 s=0

since 0 < 5 < A ; ^ n - r - l , while, if k = n - r - 1 one gets

E\|)(\(r>'I:lE(>.•)(t^M,,MJ=l j=0 s=0

=^\-iy=^)rw^ DLEMMA 4.9. — Let )% &e ê ^êr right (n - 1) x (n - 1) MocA; o/W.

, 7 . . ' • - ' Y / J ' If 7

â^ 25,

(__ ^1 ^72 . . . 77n-l

W,= .: : : :

^-2) ^-2) ... ^-l2^

/ / l^=l, ênH^=(-i)^2^-^^ ^herem^= j (n - 2)(n - 1).

PTW/. — From Lemma 4.8 we have that H^)% is upper triangularalong the tranverse diagonal. Hence, W^ equals W^ times the productof the elements on the transverse diagonal. That is

W^ = W^W^ (-1)^ = (-i)-2^-^

m-z as in the statement. []


388 G. MARI BEFFA

Also, we have the following expected result

LEMMA 4.10. — Let B be defined as in (4.7). Then B = (-l)71-1^1

Proof. — Following the notation in (4.7), the {k^j + 1) entry in B isgiven by

Qk+ij+i - 0^1.7+1 = (-i)7711 [c^+i^+i - 0^^7+1, iLwhere, in general, I denote the cofactor (z,j) of the Wronskian of the

. This entrvector function ^ by C " . This entry can be rewritten as

r]o + (f)krjk^o+

^+

(n-l) ,% +

^k^]k

W

^^"-1)

111

^)

^-1)

... ^

r^• • • ^

-(n-l)• • • %

• • • r]n-i

^)• • • ln-1

^-1)• • • 'ln-1

where, again, " signs the deletion of the term. Using the fact that all rjjcan be expressed in terms of cofactors of W^, one can easily prove that

n-lrjo = — ^ <piT]i- This fact, together with Lemma 4.8, implies that deter-

minant (4.11) has, in fact, zeros along the first column, except for the last

entry, which is equal to ̂ (^(-l)7'^ • Since n^ {n~,l)(-^)r = 0,r=0 r=0

we get that (4.11) equals

(_l)2n+l(_l)m^_^+J^ ^')

m

^

^-2)

rfk

•'- ^

.'.. ^-2)

• • • rfn-1

^)'/n-l

(n-l)fi' • • ln-1

The use of the result of Lemma 4.9 leads as to the conclusion of theproof. []

At last we can analyze the shape of M.. We can directly obtain it fromthe relationship (4.9), rewritten as

M = (-l^-^y^H^Id+A).

TOME 127 — 1999 — N° 3


We know from Lemma 4.8 that )%H^ is lower triangular along thetransverse diagonal. Since Id+A is upper triangular along the maindiagonal, we have that M must be lower triangular along the transversediagonal, as conjectured (notice that R~1 = R is a diagonal matrix).

Furthermore, one can calculate the entries of the diagonal strictly belowthe transverse diagonal of M. Let (s^n-i ̂ -2 ... Sn-i2) be the diagonalstrictly below the transverse diagonal in }%Wj\ Using Lemma 4.8 onecan find explicitly Skn-k+i, for all k.

First of all, ^ ^n-l)^ = W^ and so

E^^^^^^-E^-j=i j=iBut, since rj, = C^ = W^^C^ (again, Q, is indicating thecofactor of the Wronskian matrix of the superindex), we have that

n-l

T,^\,=W^W^j=i

and so

w^—"-^-^^.Assume that

^rr)r,?=Wn-^-W-^W,.

Again, differentiating the relationship in Lemma 4.8

E1^-1^^-!)^j=i

we can easily see that the same relationship holds for r + 1. Therefore

S.n-^ = E ̂ -k+l)^ = (-l)'-^-±tl ̂ ̂ .


390 G. MARI BEFFA

In [4], we found that

(T)._, k+iw,^=——^p, =——^-^.

Finally, ignoring the action of the matrix R, the diagonal in M strictlybelow the transverse diagonal is given by

(_1)A+1^ rn,kn-k+l + Skn-k+l

= (-D^ n-^1 ̂ + (-i)-̂ ±l w-,^ w, = o,

concluding the proof of the fact that the shape of the matrix M is theone conjectured in [4].

To finish the paper I will point out at some of the advantages of regar-ding AGD evolutions as evolutions of basic differential invariants. Apartfrom the obvious connection between two apparently unrelated subjects,there are some extra advantages about viewing the AGD evolution in theterrain of differential invariants. Namely, it opens a road to generaliza-tions of Schwarz derivatives and KdV evolutions to the case of severalindependent variables. These generalizations might have a strong rele-vance in various subjects such as the theory of solitons, inverse scatteringand other topological branches. In fact, in [8] we found and classified acomplete set of basic differential invariants for projective surfaces. Wealso described the generalizations of the Schwarz derivative to two inde-pendent variables and the generalization of KdV to this same case. Ofcourse, these generalizations assume that the SL(?2,IR) symmetry is alsoheld in the generalized case. It is not clear yet what connection this mighthave with the Poisson Geometry of PDEs and the Theory of integrablesystems.

BIBLIOGRAPHY

[1] ADLER(M.). — On a, trace functional for formal pseudo-differentialoperators and the symplectic structure of the KdV, Inventiones Math.,t. 50,1Q79, p.219-48.

TOME 127 — 1999 — ?3


[2] DRINFEL'D (V.G.), SOKOLOV (V.V.). — Lie algebras and equations ofKdVtype, J. Soviet Math., t. 30, 1985, p. 1975-2036.

[3] GEL'FAND (I.M.), DIKII (L.A.). — A family of Hamiltonian structuresconnected with integrable nonlinear differential equations. — I.M.GePfand collected papers I, Springer-Verlag, New York, 1987.

[4] GONZALEZ-LOPEZ (A.), HEREDERO (R.), MARI BEFFA(G.). — Inva-riant differential equations and the AdIer-GePfand-Dikii bracket,J. Math. Physics, to appear.

[5] INCE (E.L.). — Ordinary Differential Equations. — Longmans Green,London,1926.

[6] KUPERSHMIDT (B.A.), WILSON, (G.). — Modifying Lax equations andthe second Hamiltonian structure, Inventiones Math., t. 62, 1981,p. 403-36.

[7] MCINTOSH(L). — SL(n + ^-invariant equations which reduce toequations of Korteweg-de Vries type, Proc. Royal Soc. Edinburgh,t. 115A, 1990, p. 367-81.

[8] MAPI BEFFA (G.), OLVER (P.). — Differential Invariants for parame-trized protective surfaces, Comm. in Analysis and Geom., to appear.

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gloria marÍ beffa · resume. la theorie des invariants differentiels et les evolutions...

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